Under which conditions on $C$ non-trivial bounds on $\epsilon_1(n)$ exist? Same question for functions of multiple variables. E.g., one can derive such bounds for functions with bounded $n$-th derivative as discussed below from bounds on $\epsilon_\infty(n)$. How to find literature specifically on bounds on $\epsilon_{1}(n)$, if it exists?

For $L_\infty$-error $\epsilon_{\infty}(n)$ non-trivial such bounds can be achieved for the class of functions with bounded $n$-th derivative by using Chebyshev polynomials.

P.S. Similar results in theoretical computer science are known as ``learning of $C$ under $L_1$-error'', but the focus there is mostly on discrete domains.